State the Remainder Theorem.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the remainder when $x^3 - 5x - 7$ is divided by $x+1$, we can use the Remainder Theorem.
Step 1: State the Remainder Theorem.
The Remainder Theorem states that if a polynomial $P(x)$ is divided by $x-c$, the remainder is $P(c)$.
Step 2: Identify the polynomial $P(x)$ and the value of $c$.
The given polynomial is $P(x) = x^3 - 5x - 7$.
The divisor is $x+1$. To find $c$, we set the divisor to zero:
$x+1 = 0$
$x = -1$
So, $c = -1$.
Step 3: Substitute $c = -1$ into the polynomial $P(x)$.
The remainder is $P(-1)$.
$$P(-1) = (-1)^3 - 5(-1) - 7$$
Step 4: Calculate the value.
$$P(-1) = -1 - (-5) - 7$$
$$P(-1) = -1 + 5 - 7$$
$$P(-1) = 4 - 7$$
$$P(-1) = -3$$
The remainder is $\boxed{\text{-3}}$.
